/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 95 Differentiate. $$ y=\frac{e^... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 95

Differentiate. $$ y=\frac{e^{x}}{x^{2}+1} $$

Short Answer

Expert verified
The derivative is \( y' = \frac{e^x(x^2 - 2x + 1)}{(x^2 + 1)^2} \).

Step by step solution

01

Recognize the Rule to Apply

The function \( y=\frac{e^{x}}{x^{2}+1} \) is a quotient of two functions. To differentiate this function, we'll apply the Quotient Rule.
02

State the Quotient Rule

The Quotient Rule states that for two functions \( u(x) \) and \( v(x) \), the derivative of their quotient \( \frac{u}{v} \) is given by: \[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]In this case, let \( u = e^x \) and \( v = x^2 + 1 \).
03

Differentiate the Numerator

Differentiate the numerator function \( u(x) = e^x \). The derivative of \( e^x \) is \( e^x \), so \( u' = e^x \).
04

Differentiate the Denominator

Differentiate the denominator function \( v(x) = x^2 + 1 \). The derivative of \( x^2 \) is \( 2x \) and the derivative of constant 1 is 0, so \( v' = 2x \).
05

Plug Values into the Quotient Rule

Using the Quotient Rule: \[ y' = \frac{(e^x)(x^2 + 1) - (e^x)(2x)}{(x^2 + 1)^2} \]
06

Simplify the Expression

Simplify the expression obtained:\[ (e^x)(x^2 + 1) - (e^x)(2x) = e^x(x^2 + 1 - 2x) = e^x(x^2 - 2x + 1) \]So, \[ y' = \frac{e^x(x^2 - 2x + 1)}{(x^2 + 1)^2} \]
07

Write the Final Derivative

The derivative of the function \( y = \frac{e^x}{x^2 + 1} \) is:\[ y' = \frac{e^x(x^2 - 2x + 1)}{(x^2 + 1)^2} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quotient Rule
The Quotient Rule is a method for differentiating a function represented as the division of two other functions. When you are dealing with a function like \( y = \frac{e^x}{x^2 + 1} \), where one function is divided by another, you employ the Quotient Rule. This rule is essential because it allows us to find the derivative of the whole expression efficiently.The Quotient Rule formula is given by:
  • If \( u(x) \) is the numerator and \( v(x) \) is the denominator, then the derivative of the division is:
\[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]This formula tells us to:
  • Multiply the derivative of the numerator by the denominator.
  • Subtract the product of the numerator and the derivative of the denominator.
  • All of this is then divided by the square of the denominator.
All these steps help us focus on finding the exact slope of the function at any given point.
Exponential Functions
Exponential functions, such as \( e^x \), are unique mathematical expressions where the variable appears in the exponent. They are highly significant in calculus, especially due to their natural growth and decay processes in real-world scenarios. For example, \( e^x \) is the base of natural logarithms and has a distinct property: its derivative is itself (\( e^x \)).This property simplifies differentiation: when you see \( e^x \) in a function like the one in the original exercise, you know:
  • The derivative of \( e^x \) is simply \( e^x \).
  • No complex changes are required to determine \( u' \) when you apply the Quotient Rule.
Through this straightforward rule, exponential functions become easier to handle within differentiation and integration processes.
Derivatives of Rational Functions
Rational functions are expressions where a polynomial is divided by another polynomial. Deriving these functions can sometimes seem complex, but rules like the Quotient Rule streamline the process. In our exercise, the function is divided into two parts: \( e^x \) as the numerator and \( x^2 + 1 \) as the denominator.To differentiate rational functions effectively:
  • Identify each part of the function, labeling them as numerator \( u(x) \) and denominator \( v(x) \).
  • Calculate the derivative of each part individually. Here, \( u' = e^x \) and \( v' = 2x \).
  • Apply the Quotient Rule to combine these derivatives into the neat formula provided:
\[ y' = \frac{e^x(x^2 - 2x + 1)}{(x^2 + 1)^2} \]By breaking down these steps, calculating the derivative of a rational function becomes manageable and methodical. This process emphasizes the importance of organizing each component separately before combining them into the final derivative form.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The power supply of a satellite is a radioisotope (radioactive substance). The power output \(P,\) in watts \((W),\) decreases at a rate proportional to the amount present and \(P\) is given by $$P=50 e^{-0.004 t}$$ where \(t\) is the time, in days. a) How much power will be available after 375 days? b) What is the half-life of the power supply? c) The satellite cannot operate on less than \(10 \mathrm{~W}\) of power. How long can the satellite stay in operation? d) How much power did the satellite have to begin with? e) Find the rate of change of the power output, and interpret its meaning.

Differentiate. $$ f(x)=e^{x / 2} \cdot \sqrt{x-1} $$

Graph $$ f(x)=\left(1+\frac{1}{x}\right)^{x} $$ Use the TABLE feature and very large values of \(x\) to confirm that \(e\) is approached as a limit.

Differentiate. $$ y=\frac{e^{3 t}-e^{7 t}}{e^{4 t}} $$

As part of a study, students in a psychology class took a final exam. They took equivalent forms of the exam at monthly intervals thereafter. After \(t\) months, the average score \(S(t),\) as a percentage, was found to be given by $$ S(t)=78-15 \ln (t+1), \quad t \geq 0 $$. a) What was the average score when they initially took the test, \(t=0 ?\) b) What was the average score after 4 months? c) What was the average score after 24 months? d) What percentage of their original answers did the students retain after 2 years ( 24 months)? e) Find \(S^{\prime}(t)\). f) Find the maximum and minimum values, if they exist. g) Find \(\lim _{t \rightarrow \infty} S(t)\) and discuss its meaning.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.